Dunlap

Tobiason J D, Dunlap J R and Rohifing E A 1995 The unimolecular dissociation of HCO a spectroscopic study of resonance energies and widths J. Cham. Phys. 103 1448-69... 

Dunlap B I 1987 Symmetry and degeneracy in Xa and density functional theory Advances in Chemicai Physics vol LXIX, ed K P Lawley (New York Wiley-Interscience) pp 287-318... 

Acknowledgements—This work was supported by the Office of Naval Research (ONR) through the Naval Research Laboratory and directly through the ONR (Chemistry-Physics and Materials Divisions. We thank D. H. Robertson, D. W. Brenner, and B. 1. Dunlap for many useful discussions. 

Fig. 2. Optimized toroidal structures of Dunlap s tori (a) torus C540 and (b) torus 05, pentagons and heptagons are shaded.

Geometric construction of tori. Possible constructions of tori with pentagons, heptagons, and hexagons of carbon atoms are given independently by Dunlap[10], Chernozatonskii[l 1], and us[12-17]. In ref. [18], the method-of-development map was used to define various structures of tori. For other tori, see ref. [19]. 

The properties of optimized helical structures, which were derived from torus C54D and Cs7a, >yps (A), (proposed by Dunlap) and torus C ,o> Dpe (B), (proposed by us) by molecular dynamics were compared. (see Figs. 9 (a) and 10). (Although the torus Cs7f, is thermodynamically stable, helix 57 was found to be thermodynamically unstable 14]. Hereafter, we use helix C to denote a helix consisting of one torus (C ) in one pitch. 

In this paper we elaborate models of perfect tubule connections leading to curved nanotubes, tori or coils using the heptagon-pentagon construction of Dunlap[ 12,13]. In order to understand the mechanisms of formation of perfectly graphitized multilayered nanotubes, models of concentric tubules at distances close to the characteristic graphite distance and with various types of knee were built. (Hereafter, for the sake of clarity, tubules will be reserved to the individual concentric layers in a multilayered nanotube.)... 

Dunlap describes ihe connection between (L,0) and (L, L tubules by means of knees. A knee is formed by the presence of a pentagon on the convex and of a heptagon on the concave side of the knee. An example is illustrated in Fig. 3(a). The (12,0)-(7,7) knee is chosen for illustration because it connects two tubules whose diameters differ by only 1%. The bent tubule obtained by that connection was called ideal by Dunlap[ 12,13]. 

As distinct from ihe ideal connection of Dunlap, we now describe the series of nanotubule knees (9 ,0)-(5m,5 ), with n an integer. We call this series the perfectly graphitizahle carbon nanotuhules because the difference of diameter between the two connected segments of each knee is constant for all knees of the series (Fig. 4). The two straight tubules connected to form the = 1 knee of that series are directly related to Cfio, the most perfect fullerene[15], as shown by the fact that the (9,0) tubule can be closed by 1/2 Qo cut at the equatorial plane perpendicular to its threefold rotation symmetry axis, while the (5,5) tubule can be closed by 1/2 Qo cut at the equatorial plane perpendicular to its fivefold rotation symmetry axis [Fig. 5(a)]. 

B. Dunlap, In FuHerenes Recent Advances in the Chemistry and Physics of FuHerenes and Related Materials, (Edited by K. Radish and R. S. Ruoff), p. 226. The Electrochemical Society, Pennington, NJ (1994). 

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